According to the laws of Shariya, a woman should put on a headcover at home or not? (in front of the family only: husband, sons, and daughters).
No, the husband, sons, brothers etc. of a woman are her mahrams. She does not need to cover her head in front of them.
قل للمؤمنات ... لا يبدين زينتهن إلا لبعولتهن أو آبائهن أو آباء بعولتهن أو أبنائهن أو أبناء بعولتهن أو إخوانهن أو بني إخوانهن أو بني أخواتهن أو نسائهن أو ما ملكت أيمانهن أو التابعين غير أولي الإربة من الرجال أو الطفل الذين لم يظهروا على عورات النساء
And tell the believing women to ... not expose their adornment except to their husbands, their fathers, their husbands' fathers, their sons, their husbands' sons, their brothers, their brothers' sons, their sisters' sons, their women, that which their right hands possess [i.e., slaves], or those male attendants having no physical desire, or children who are not yet aware of the private aspects of women.
This permits her to expose her adornments (and hence the parts of the body which are adorned) to her mahrams. These parts include at least the head, face, hands and feet as they are adorned with tiaras, necklaces, cosmetics, rings and anklets.
ويحوز للرجل أن ينظر من ذوات محارمه إلى ما يظهر غالبا كالرقبة والرأس والكفين والقدمين ونحو ذلك
I'll add however that this is with regard to covering because of people. A woman is still required to cover her head during salah, even if she is at home and only her family is there or even if she is alone.
لا يقبل الله صلاة حائض إلا بخمار
Salat of a women who has reached the age of menstruation is not accepted without a Khimar
NOTE: This is my own answer based upon my own translations and research, before any moderator wishes to delete it.
It is not obligatory for a woman to cover her head at home before Maĥrams, however it is good to do so out of modesty.
Imām Sayyid Muĥammad Amīn Ibn Áābidīn al-Ĥanafī al-Shāmī [1198-1252 AH / 1784-1836 CE] writes in Radd al-Muĥtār, quoting Qunyah al-Munyah:
“It is best for her to wear a thin Khimār, that describes what is under it, in the presence of her Maĥrams.”
This is also quoted in Fatāwā al-Hindiyyah:
Ĥakīm al-Ummah Muftī Aĥmad Yār Khān al-Ĥanafī al-Naýīmī [d. 1391 AH / 1971 CE] writes in his Tafsīr:
Therefore it can be observed that though the scholars did not consider it obligatory for a woman to cover her head in the presence of her Maĥrams, however they still considered it superior and more appropriate.
A woman is not obliged to cover her hair in front of her husband, sons, daughters etc. (Reference: Surah An-Nur 24:31). There is neither thawab for covering her hair nor sin for uncovering it in the presence of her husband and Mahrams.
The assertions by ulemas of Indian subcontinent, that a woman should cover her hair at home or during adhan, is non-sense and has no basis in Quran and Sunnah.